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On the Conformal Geometry of Transverse Riemann-Lorentz Manifolds

On the Conformal Geometry of Transverse Riemann-Lorentz Manifolds. V. Fernández Universidad Complutense de Madrid. Transverse Riemann-Lorentz Manifolds. M connected manifold symmetric (0,2) tensor field on M. ∑:={ p M : degenerates }≠ ø Rad p (M):={ : } ≠ø, p ∑

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On the Conformal Geometry of Transverse Riemann-Lorentz Manifolds

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  1. On the Conformal Geometry of Transverse Riemann-Lorentz Manifolds V. Fernández Universidad Complutense de Madrid

  2. Transverse Riemann-Lorentz Manifolds M connected manifold symmetric (0,2) tensor field on M ∑:={ pM : degenerates}≠ø Radp(M):={ : }≠ø, p∑ M-∑ is a union of pseudoriemannian manifolds

  3. DEF: M is a transverse type-changing manifold if for every p∑ ∑ is a hypersurface that locally separates two open pseudoriemannian manifolds whose indices differ in one and dim(Radp(M))=1

  4. DEF: M is a transverse Riemann-Lorentz manifold if the components of M-∑ are either riemannian or lorentzian. Example: ∑ : type-changing hypersurface M+: riemannian M-: lorentzian

  5. On M-∑ we have all the geometric objects associated to g derived from the Levi-Civita connection D: covariant derivatives, parallel transport, geodesics, curvatures… PROBLEM: Extendability to M of these objects?

  6. □ the unique torsion-free and metric dual connection on Msuch that on M-∑ R a radical vectorfield (RpRadp(M)-{0}) well-defined map

  7. DEF: M is II-FLAT if OBS: Mis II-flat iff extends to M whenever or are tangent to ∑. DEF: M is III-FLAT if

  8. THEOREM: (Kossowski,97) K covariant curvature extends to M iff radical transverse to ∑ and M II-flat. Ric Ricci tensor extends to M iff radical transverse to ∑ and M III-flat.

  9. Conformal Geometry transverse Riemann-Lorentz C= conformal structure is also transverse Riemann-Lorentz whith the same type-changing hypersurface ∑ and same radical Rad DEF: conformal transverse Riemann-Lorentz manifold

  10. Weyl Conformal Curvature DEF: N pseudoriemannian manifold Weyl tensor of N, where Schouten tensor Kulkarni-Nomizu product

  11. OBS: thus conformal invariant, called Weyl conformal curvature THEOREM: (Weyl, 1918) iff Mconformally flat that is, around every pN there exists a metric on the conformal structure which is flat

  12. Extendability of Weyl tensor THEOREM: W extends to the whole M iff radical transverse and M conformally III-flat that is, around every p∑ there exists a metric on the conformal structure which is III-flat

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